No one who lets their cats outside to roam freely can guarantee where they do and don't go. Cats allowed outside will roam and sometimes farther then the owners have any clue about. I live out in the country and shockingly have no strays that live in my neighborhood. My neighbor behind me lets her cats out but they tend to stay off our property because our dog would chase them. In all the years I have lived here I have only had one stray show up and she was a kitten and got added to my current crew. She has no desire at all to go out. My other cats are strictly indoors and the only outside time they ever get is strictly supervised by me.
Do you know if your area has any kind of leash law or ordinance about free roaming cats? Not many do but if yours does at least you have the law on your side. Regarding the sign on his property, not sure if there is anything you can do about that either as it is his property and even though the sign is rude and intimidating the wording is not libelous. It is unfortunate that the situation has escalated to this. I am sorry you are having to go through this. As a cat lover I would be very upset if neighborhood cats were frequenting my property and using it as a litter box. They can also endanger my house cats by spreading disease. I don't think I would have handled it any differently then you did. You spoke to your neighbor prior to trapping and he told you they didn't go on your property. Obviously they did. Ultimately, if your nieghbor does not want his cats trapped and picked up by animal control, or hit by a car, or mauled by a stray dog he should make them inside cats. It is his responsibility, not yours, to keep his cats safe and under control. You could try speaking to your neighbor again, reminding him you asked if his cats were coming on to your property and he said no so you thought you were trapping stray cats that could be spreading disease. Good luck, I know it is awful to be at odds with your neighbors.